On Distance in Some Finite Planes and Graphs Arising from Those Planes

I Dogan,A Akpinar,Ahmet Sinan Cevik

doi:10.1155/2021/6668682

Abstract

In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

Highlights

Field planes are a class of projective planes, and the majority of projective planes are not in this class. e known finite projective planes are of the order pr
Our second aim is to examine the geometric correspondences of circles, ellipses, hyperbolas, and parabolas, which are basic geometric concepts whose properties are well known from Euclidean geometry, in the finite planes. e aim of this study is to obtain some numerical results, such as how many different or identical circles there are with a radius of 1, 2, 3, and so on. is numerical examination will be limited to the projective planes of orders 2, 3, 4, 5, 7, and 8, which are known to be unique, to the field projective plane of order 9, and, by a generalization from this point, to the projective planes of order pr
Based on the following algebraic structure and some information to be given about the class of octonion planes constructed with this algebraic structure, we will give a definition of distance that is not encountered in the literature on finite projective planes or finite projective Klingenberg planes

Summary

Introduction

E correspondence of such a line in graph language can be considered as a path In this case, finite PK planes (and finite projective planes and finite affine planes) become appropriate geometric models for solving some important problems in graph theory. Our second aim is to examine the geometric correspondences of circles, ellipses, hyperbolas, and parabolas, which are basic geometric concepts whose properties are well known from Euclidean geometry, in the finite planes. The roles of finite affine and projective Klingenberg planes for all the results obtained are considered and their equivalences in graph applications are discussed

Preliminaries

The Definition of Distance in a PK Plane

The Main Results

Conclusions